Lemma 3.23 (Proposition 4.1, [1]}; Proposition 2, [2]}; Section 19.4, [3]}) 
Let \(K_1, \ldots , K_n\)  be some convex bodies of \(\mathbb {R}^n\)  and suppose that \(K_{n-m+1, \ldots , K_n}\)  are contained in some \(m\) -dimensional affine subspace \(U\)  of \(\mathbb {R}^n\) .
Let \(MV_U\)  denote the mixed volume with respect to the \(m\) -dimensional volume measure on \(U\) , and let \(MV_{U^\perp }\)  be defined similarly with respect to the orthogonal complement \(U^\perp \)  of \(U\) .
Then the mixed volume of \(K_1, \ldots , K_n\)
\(&MV(K_1, \ldots , K_{n-m}, K_{n-m+1}, \ldots , K_n)\\&= \binom{n}{m}^{-1} MV_{U^\perp } (K_1^{\prime }, \ldots , K_{n-m}^{\prime })MV_U(K_{n-m+1, \ldots , K_n),}\) 
